‘Not a work of obvious bling’: the Garage Museum of Contemporary Art in Moscow’s Gorky Park. Photograph: Yuri Palmin

The coincidence, possibly to the annoyance of those involved, is too great to ignore: two new museums of contemporary art in two great European cities, both created by women of private wealth, both involving the making-over of existing buildings of un-obvious charm, and both designed by Rem Koolhaas and his practice, OMA. It suggests we are seeing the future or, rather, the present of cultural building. As public backing recedes it falls more to individual patrons, which puts such museums at the mercy of their benevolence and intelligence.

The newest of the two is the Garage Museum of Contemporary Art, which opened last Friday in Gorky Park, Moscow. It is the creation of Dasha Zhukova, the Russian-born and Californian-raised art collector, and the derelict Seasons of the Year restaurant, a Brezhnev-era public canteen that once served many hundreds of people at a sitting.

As Zhukova is married to Roman Abramovich, who has contributed to an unknown degree to the cost of the Garage, the place is a diagram of the switches of power and money in the past few decades of Russian history, from Soviet mass catering to last week’s elegant opening party, but it is not an oligarchic treasury, or a work of obvious bling. It presents a box wrapped in polycarbonate – translucent plastic – to the outside world, while the interior features battered concrete and erratic green tiling from the original building. Two huge panels of polycarbonate, one on either flank of the box, slide upwards like inverse portcullises to announce its accessibility. (Although, disappointingly, they reveal a glass screen more than the actual interior.)

Rem Koolhaas likes to stress how important Russia is to him. On a visit to Moscow in 1965, he saw the drawings of constructivists such as Ivan Leonidov who showed how architecture can “propose different ways in which you might live”. The experience persuaded Koolhaas to change profession, from writer to architect. He also likes to tell how, in the past few years, OMA has become interested in the preservation of old buildings, an apparently surprising switch for a practice inspired by constructivists and dedicated to the contemporary.

At a personal level, Koolhaas says this interest is about how you manage a career. “Becoming well-known involves a huge lack of control. There are expectations that you have to outwit.” Working with old buildings means “you can practise your profession without expectations of extravagance”; it is “a tactic to avoid the pressure to create a series of singular masterpieces”. In terms of wider culture, he notes how the rise of conservation coincided with the rise of modernist architecture in the late 19th and 20th centuries, and that the two are therefore intertwined: “if you want to change the world you also have to decide what you want to keep.”

He says that conservation movements are too often preoccupied with the beautiful at the expense of the everyday. He also says that “you can say so many things about the Soviet system that were bad, but in terms of public architecture it was generous”. Together, these views lead to the main and best idea of the Garage, which is to keep the frame of the old building, its mosaics and tiles, and a central staircase of floating terrazzo treads, of unfeasible width and some grace, up which the proletarian lunchers might ascend as if to a palace ballroom. The attrition of history is retained, in a less feverish version of the approach that David Chipperfield used on the war-damaged Neues Museum in Berlin.

Dasha Zhukova, founder of the Garage Museum of Contemporary Art, with its architect, Rem Koolhaas, at the opening of the museum. Photograph: Pavel Golovkin/AP

There are dangers here – making a fetish of damage, or creating the sort of Soviet chic that, in swanky Moscow restaurants, leads to huge red stars being placed against backgrounds of gold leaf – but they are avoided. The project has few of the tics and the stunts that OMA often likes to pull. “This building doesn’t introduce irony or distancing humour,” says Koolhaas. “It’s in your face, it’s there.” He’s not wrong. The main entrance space, with the stairs facing the mosaic and openings at either end to the park, is indeed generous. If modern Russian history has proceeded by a series of violent erasures of preceding politics and architecture, here the encounter of the contemporary and the Soviet is at least stated.

What gets lost, for now at least, is the art. Some of the exhibition spaces are squeezed around the periphery of the central core that used to contain the kitchens. Others are wide open decks that require a high degree of curatorial oomph to occupy. It doesn’t help that the building has been finished in a hurry – indeed it is not really finished now – with consequences for the quality of detail. The exhibition programme has nice ideas, in particular a desire to reveal overlooked strands of art in postwar Russia, through such things as personal photographs of avant-garde artists in the dark years of the 70s and 80s, and an exploration of the film-making that came out of cultural exchanges with Africa, but the execution is sketchy.

It is refreshing not to be greeted at a new arts centre with a glossy chunk of Kapoor or Koons. There is also an exhibition about the American National Exhibition held in Moscow in 1959, a proclamation of technological prowess, consumer satisfaction and artistic freedom expressed through Pollock, O’Keeffe, Eames, Kodak and Pepsi, and refrigerators and geodesic domes, which prompted the famous kitchen debate between First Secretary Khrushchev and Vice-President Nixon. This is a terrific subject for a show, which could have filled the Garage, but this one feels rushed and provisional.

A large outdoor area for events is to be completed. Yet to be started is the restoration of a nearby 1920s exhibition pavilion, a large and celebrated structure called the Hexagon. When this is filled with galleries, and when the programme rises to the challenges set by the architecture, the art might begin to exceed its current role, which is to furnish incidents in a rewarding sequence of publicly accessible spaces.

Fondazione Prada, in Milan, also awaits the completion of a significant amount of gallery space, in a concrete tower that is still under construction, but otherwise is looking smoother than the Garage. Budgets are not disclosed, but this, the conversion of an old distillery into a venue to display the collection of Miuccia Prada and Patrizio Bertelli, seems to have cost much more than the £20m-plus rumoured for the Moscow project.

Here, as Koolhaas says, the existing array of sheds would have been “really dull” without some decisive interventions, so OMA has made them into a set of courts interrupted by such things as a mirror-walled auditorium block, a glass-walled, travertine-floored gallery and an existing tower covered in gold leaf. The side walls of the auditorium can pivot upwards, to allow a flow of space from one side of the complex to the other. Koolhaas points out that the task here was to make many things into a single entity, whereas the Garage is a case of bringing multiplicity to something already entire.

As in a film by Antonioni, there are appearances and disappearances, suggestive vacancies, mirages. Shadow and reflection are as important as material substance. The courts suggest a shared communal life of the kind for which Italy is famous, and to a degree it is there, but the architecture also frames views of other spectators that makes them into objects of display, and at other times offers a disquieting blankness, in ways that, again as in Antonioni, suggest both connection and alienation.

It is mesmerising and intensely considered and self-conscious. Like the Garage, it challenges the installers of art, but in a different way. It offers a range of types of places for display, from the glass-walled gallery to simple cells in the existing structures, external areas, to the rooms in the forthcoming tower to industrial environments, which, says Koolhaas, aim to outdo what he calls the “post-Serota” tendency to put art in formerly working structures. While presenting opportunities to curators, this range obliges them to negotiate jump-cuts and disconnections. Curators also have to deal with the high degree of conceptualisation in the conceiving of the place: everything, architecture and art, feels like a concept. It is rarely allowed just to be.

The most obvious issue, for both projects, is the relation of private money to public good. Both resemble the cultural gifts of JP Morgan or Andrew Carnegie in the American Gilded Age, where colossal wealth was recycled in a way that combined personal motives with real common benefit. Fondazione Prada, as a converted factory with a new tower, also invites comparison with the similarly configured SESC Pompéia in Sao Paulo, a cultural, social and leisure centre designed by Lina Bo Bardi that, by putting everyday activities alongside art, made a truly populist and popular institution.

The Accademia dei bambini at Fondazione Prada, Milan, has been designed for children. Photograph: PR

Prada and Bertelli aren’t claiming to offer a SESC. It is still very much their territory, a manifestation of their rarefied cultural world, which the public is invited to enter and in which we are to some degree encouraged to take part. Which is achieved with considerably more charm and brilliance than, for example, the current version of the Saatchi collection in London. Of the two Koolhaas projects, the Garage promises to be more fully engaging, being easily accessible from Moscow’s most important park – a statement made with caution, given the ease with which apparent public gains can be reversed in Russia.

But what if high-value art were not the main medium of munificence? What if the architectural ingenuity lavished in such projects went on something like the affordable canteen that the Garage once was? “I am convinced that this is needed,” says Koolhaas of the Garage’s artistic function, but “it is a tragedy of the modern world that architects like me are never asked to design something like a canteen”

The Guggenheim’s proposed Finnish outpost is one of the biggest architectural competitions ever, but it has been criticised as a vanity project. Now, a counter-competition has been launched in rebellion